Geometry of secret sharing

The Shamir secret sharing scheme uses univariate interpolation to recover an unknown value of a polynomial (the secret) from known values (the shares). The scheme generalizes in various directions. We discuss the scheme in the setting of graphs and coding theory, and in the setting of multivariate polynomials and algebraic functions. Schemes defined using curves are particularly important, both for their asymptotic performance and for their application to secure multi-party computation. We show that properties of such schemes are often much better than expected and we give best known bounds for their parameters (joint work with Seungkook Park and Radoslav Kirov; see also http://agtables.appspot.com).